$A$ $4 \ kg$ stone attached at the end of a steel wire is being whirled at a constant speed $12 \ ms^{-1}$ in a horizontal circle. The wire is $4 \ m$ long with a diameter $2.0 \ mm$ and Young's modulus of the steel is $2 \times 10^{11} \ Nm^{-2}$. The strain in the wire is:

Two masses $m_1$ and $m_2$ are connected by a light string passing over a pulley as shown in the figure. If the cross-sectional area of the string is $S$,its length is $L$,and its Young's modulus is $Y$,what is the elongation in the string?

Four identical hollow cylindrical columns of mild steel support a big structure of mass $50,000 \; kg$. The inner and outer radii of each column are $30 \; cm$ and $60 \; cm$ respectively. Assuming the load distribution to be uniform,calculate the compressional strain of each column. (Take Young's modulus of mild steel $Y = 2.0 \times 10^{11} \; Pa$ and $g = 9.8 \; m/s^2$)

$A$ horizontal steel railroad track has a length of $100 \, m$ when the temperature is $25^{\circ} C$. The track is constrained from expanding or bending. The stress on the track on a hot summer day,when the temperature is $40^{\circ} C$,is ............. $\times 10^7 \, Pa$. (Note: The linear coefficient of thermal expansion for steel is $1.1 \times 10^{-5} /^{\circ} C$ and the Young's modulus of steel is $2 \times 10^{11} \, Pa$.)

When a certain weight is suspended from a long uniform wire,its length increases by $1 \ cm$. If the same weight is suspended from another wire of the same material and length but having a diameter half of the first one,then the increase in length will be ........ $cm$.

$A$ compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross-section, $Y$ its Young's modulus, and $\alpha$ its coefficient of linear expansion. Then, $F$ is equal to